A direct application of the Kato-Rellich theorem yields the essential
self-adjointness of the (massive, 3d) Dirac-Coulomb Hamiltonian only for
\(|\lambda |<\frac{1}{2}\) . This proof is due to Kato
[1]}, [2]}. The essential self-adjointness up to the boundary
of the “regular region”
\(|\lambda |<\frac{\sqrt{3}}{2}\) 
was proven independently by Gustaffson-Rejtö [3]}, [4]} and Schmincke
[5]}. They needed to use slightly more refined
arguments going beyond to the basic Kato-Rellich theorem. The “distinguished self-adjoint extension” in the region
\(\frac{\sqrt{3}}{2}<|\lambda |<1\)  was described in several equivalent ways,
mostly involving the characterization of the domain, by Schmincke,
Wüst, Klaus, Nenciu and others [6]}, [7]}, [8]}, [9]}.
Esteban and Loss [10]}
characterized the distinguished self-adjoint realization at the boundary
of the “transitory region”, that is for \(|\lambda |=1\) , by using
the so-called Hardy-Dirac inequalities. Self-adjoint realizations in the
“supercritical region” \(|\lambda |>1\)  were studied by Hogreve in
[11]}.
